Ultimate Limit State: Bending

Transcription

1 Ultimate Limit State: Bending Presented by: John Robberts Design Point Consulting Engineers (Pty) Ltd

2 ULS: Flexure with and without axial force fundamental principles There are three fundamental principles (that has not changed) for ULS (Cl. 6.1(2)P): 1. Plane sections remain plane: Unless it is a deep beam, where the span less than 3 times the overall section depth (Cl (3)) Cracking in the tensile zone can be ignored (provided the gauge length under consideration spans more than one crack). 2. The stress-strain relationships for the materials are known: Concrete stress-strain relationships are defined in Figs. 3.2 to 3.5 Non-prestressed reinforcement stress-strain relationships are defined in Fig. 3.8 Prestressed reinforcement stress-strain relationships are defined in Fig At each section the actions (applied forces and moments) must be in equilibrium with the action effects (internal stresses). Although not explicitly defined in Cl. 6.1(2)P, it is fundamental to analysis and design.

3 Additional principles, related to materials (as before) at ULS (Cl. 6.1(2)P): Strain in bonded reinforcement or prestressing tendons is the same as that in the surrounding concrete Provided the gauge length under consideration spans more than one crack. Unbonded prestressing tendons obviously do not comply here. The tensile strength of the concrete is ignored. Initial strain in prestressing tendons needs to be taken into account. Failure at ULS occurs when: 1. The concrete reaches it s ultimate strain (e.g for concrete class C50/60). 2. The prestressed or non-prestressed reinforcement ruptures when the design strain limit (e ud ) is exceeded. This failure mode is highly unlikely if the limits to the neutral axis depth are complied with.

4 Notation and Terminology Paragraphs are numbered as: (x)p for Principles. No alternatives are allowed, unless specifically stated. (x) for Application Rules. Generally accepted methods, which follow the principles and satisfy their requirements. Alternatives are possible, provided that it will comply with the principles with regard to structural safety, serviceability and durability. Loads are now referred to more generally as Actions, to include imposed deformations (e.g. temperature and settlement). General notation: Permanent (G) = Dead Loads Variable (Q) = Live or Wind Loads Accidental (A) Prestressing (P), which is treated as a permanent action in most situations.

5 Action Effects collectively refer to the member forces, bending moments, shears and associated deformations. Characteristic values have a subscript k. Design values have a subscript d, which takes into account the relevant partial safety factors applied to the characteristic values. References to EN 1991 (Actions on Structures) should be replaced by SANS (2011). All formulae and expressions in the code refers to the cylinder strength (f ck ). When used, the cube strength will be denoted by f ck,cube (e.g. see Table 31.). The cube strength is always higher than the cylinder strength due to frictional effects at the boundaries and differences in aspect ratios of test spcimen. A concrete class C50/60 refers to a cylinder strength of 50 MPa and a cube strength of 60 MPa For cube strengths less than approximately 80 MPa, a reasonable approximation will be: f ck = 0.8 f ck,cube Quality control in South Africa (as in the UK) will continue to use cube strengths.

6 Cylinder strength (MPa) Cylinder strength fc (MPa) mm cylinders Eurocode 2 relationship between cylinder and cube strengths (Table 3.1) EC2 values Ratio = Experimental relationship between cylinder and cube strengths (Table 3.1) Cube strength (MPa) 100 mm cubes f cu Held (Ref. 2-10) Smeplas (Ref. 2-11) f = 0.8 f c cu Cube strength (MPa)

7 Design concrete compressive strength From Eurocode 2: It is recommended to follow the UK National Annex here: α cc = 0,85 for compression in flexure and axial loading. α cc = 1,0 for other phenomena. However,α cc = 0,85 may be taken conservatively for all phenomena.

8 The coefficient α cc takes into account: The long-term effects on the compressive strength. Unfavourable effects resulting from the way the load is applied. The Commentary to Eurocode 2 (Beeby et al., 2008a) justifies the use of α cc = 1,0 by taking into account that: Under a constant load, the stress at failure approaches 80% of the short-term capacity. The 28 day strength is used in design while the structure will experience the design load at a much later stage when the concrete strength has increased (about 12%). Resistance equations in a design code of practice is based on full-scale tests which occur over about 90 minutes, whereas the short-term capacity is determined in about 2 minutes. This means that the reduction in strength of about 15% is accounted for in the full-scale test. The background document to the UK National Annex (PD 6687, 2006) recommends α cc = 0,85. They point out that, at higher strengths: The Eurocode 2 value (α cc = 1,0) represents an upper characteristic value. The UK National Annex recommendation (α cc = 0,85) represents a median.

9 Background document to the UK National Annex (PD 6687, 2006). For European research also see: fib 52 (2010) Structural Concrete - Textbook on behaviour, design and performance, Volume 2, Basis of design, fib Bulletin No. 52 Fédération Internationale du Béton, Lausanne. For American research also see ACI 318 and: Rüsch, H. (1960). Researches Toward a General Flexural Theory for Structural Concrete, ACI Journal, July, Vol. 57 No. 1.

10 Concrete stress-strain relationship Four stress-strain relationships are available for concrete in compression: 1. Parabolic: for non-linear structural analysis (Fig. 3.2). 2. Parabolic-rectangular: for design of cross-sections (Fig. 3.3). 3. Bi-linear: for design of cross-sections (Fig. 3.4). 4. Equivalent rectangular: for design of cross-sections (Fig. 3.5). The equivalent rectangular stress block is recommended for general design: The parameters for the stress-strain relationships have been calibrated to yield similar results. The section will be designed for flexure so that the reinforcement yields prior to failure of the concrete. Therefore, the reinforcement stress-strain relationship dominates the behaviour and the exact shape of the concrete stress-strain relationship has minimal impact.

11 λ = 0,9 in SANS η = 0,85 in SANS 10100

12 Redistribution of moments Linear elastic analysis with limited redistribution (Cl. 5.5)

13 Moment redistribution ratio (β b in SANS 10100): moment at section after redistribution δ = moment at section before redistribution 1 For concrete Classes up to C50/60: x u δ k 1 + k 2 d From SANS C. 5.5(4): k 1 = 0,4 and k 2 = 0,6 + 0,0014 ε cu2 For concrete Classes up to C50/60: k 2 = 0,6 + 0,0014 0,0035 = 1 So that δ x u d Rearranging to have the neutral axis depth on the left, we obtain (the same equation as before): x u δ 0.4 d Also from Cl (2): x u d 0,45 Which implies δ = 0.85, or 15% redistribution of moments (In SANS β b = 0.9 or 10% redistribution of moments).

14 Design of flexural members (without axial force) The Eurocodes focus on defining design principles rather than providing design recipes. Design equation can be found in text books such as Bond et al. (2007):

15 (Bond et al. 2007)

16 (MPa)

17 Flanged beams Effective flange width (Cl ) is slightly different than before:

18 The distance l 0 between points of zero moment are defined below For the majority of flanged sections, where the flange is in compression, the depth of the stress block will fall inside the flange: 0,8 x < h f or x < 1,25h f And the equations for a rectangular section will apply, with b = b eff

19 Bending plus axial load at ULS For pure compression the strain in the concrete is limited to ε c3 if the equivalent rectangular stress block is used. For concrete classes up to C50/60 ε c3 = 0, With bending, but without tensile strain in the section, the strain pivots about a hinge point at h/2. (Bond et al. 2007)

20 References 1. Mosley, W. B.; Bungey, J. H. & Hulse, R. (2012). Reinforced Concrete Design to Eurocode 2, 7 th Ed., Palgrave Macmillan, 448 pp. 2. Narayanan, R. S. & Goodchild, C. (2006). Concise Eurocode 2: For the design of in-situ concrete framed buildings to BS EN : 2004 and its UK National Annex: 2005, MPA - The Concrete Centre, Camberley, 107 pp. 3. Bond, A. J.; Harrison, T.; Brooker, O.; Moss, R.; Narayanan, R.; Webster, R. & Harris, A. J. (2007). How to Design Concrete Structures using Eurocode 2 - The Compendium, MPA Concrete Centre, 98 pp. 4. PD 6687 (2006). Background paper to the UK National Annexes to BS EN , British Standards Institute, London, 40 pp. 5. Beeby, A. W.; Peiretti, H. C.; Walraven, J.; Westerberg, B. & Whitman, R. V. Jacobs, J.-P. (Ed.) (2008a). Eurocode 2 Commentary, European Concrete Platform ASBL, Brussels Beeby, A. W.; Peiretti, H. C.; Walraven, J.; Westerberg, B. & Whitman, R. V. Jacobs, J.-P. (Ed.) (2008b). Eurocode 2 Worked Examples, European Concrete Platform ASBL, Brussels IStructE (2010). Manual for the design of concrete building structures to Eurocode 2, The Institution of Structural Engineers, London, 246 pp. 8. IStructE (2006). Standard Method of Detailing Structural Concrete - A manual for best practice, 3 rd. Ed., The Institution of Structural Engineers, London, 188 pp. 9. Narayanan, R. S. & Beeby, A. (2005). Designers' Guide to EN and EN Eurocode 2: Design of Concrete Structures. General Rules and Rules For Buildings and Structural Fire Design, Thomas Telford, London, 218 pp. 10. Threlfall, T. (2013). Worked Examples for the Design of Concrete Structures to Eurocode 2, CRC Press, Boca Raton,243 pp.